# §18.2 General Orthogonal Polynomials

## §18.2(i) Definition

### Orthogonality on Intervals

Let $(a,b)$ be a finite or infinite open interval in $\mathbb{R}$. A system (or set) of polynomials $\{p_{n}(x)\}$, $n=0,1,2,\ldots$, is said to be orthogonal on $(a,b)$ with respect to the weight function $w(x)$ ($\geq 0$) if

 18.2.1 $\int_{a}^{b}p_{n}(x)p_{m}(x)w(x)\,\mathrm{d}x=0,$ $n\neq m$.

Here $w(x)$ is continuous or piecewise continuous or integrable, and such that $0<\int_{a}^{b}x^{2n}w(x)\,\mathrm{d}x<\infty$ for all $n$.

It is assumed throughout this chapter that for each polynomial $p_{n}(x)$ that is orthogonal on an open interval $(a,b)$ the variable $x$ is confined to the closure of $(a,b)$ unless indicated otherwise. (However, under appropriate conditions almost all equations given in the chapter can be continued analytically to various complex values of the variables.)

### Orthogonality on Finite Point Sets

Let $X$ be a finite set of distinct points on $\mathbb{R}$, or a countable infinite set of distinct points on $\mathbb{R}$, and $w_{x}$, $x\in X$, be a set of positive constants. Then a system of polynomials $\{p_{n}(x)\}$, $n=0,1,2,\ldots$, is said to be orthogonal on $X$ with respect to the weights $w_{x}$ if

 18.2.2 $\sum_{x\in X}p_{n}(x)p_{m}(x)w_{x}=0,$ $n\neq m$,

when $X$ is infinite, or

 18.2.3 $\sum_{x\in X}p_{n}(x)p_{m}(x)w_{x}=0,$ $n,m=0,1,\ldots,N;n\neq m$,

when $X$ is a finite set of $N+1$ distinct points. In the former case we also require

 18.2.4 $\sum_{x\in X}x^{2n}w_{x}<\infty,$ $n=0,1,\dots$,

whereas in the latter case the system $\{p_{n}(x)\}$ is finite: $n=0,1,\ldots,N$.

More generally than (18.2.1)–(18.2.3), $w(x)\,\mathrm{d}x$ may be replaced in (18.2.1) by a positive measure $\,\mathrm{d}\alpha(x)$, where $\alpha(x)$ is a bounded nondecreasing function on the closure of $(a,b)$ with an infinite number of points of increase, and such that $0<\int_{a}^{b}x^{2n}\,\mathrm{d}\alpha(x)<\infty$ for all $n$. See McDonald and Weiss (1999, Chapters 3, 4) and Szegő (1975, §1.4).

## §18.2(ii) $x$-Difference Operators

If the orthogonality discrete set $X$ is $\{0,1,\dots,N\}$ or $\{0,1,2,\dots\}$, then the role of the differentiation operator $\ifrac{\mathrm{d}}{\mathrm{d}x}$ in the case of classical OP’s (§18.3) is played by $\Delta_{x}$, the forward-difference operator, or by $\nabla_{x}$, the backward-difference operator; compare §18.1(i). This happens, for example, with the Hahn class OP’s (§18.20(i)).

If the orthogonality interval is $(-\infty,\infty)$ or $(0,\infty)$, then the role of $\ifrac{\mathrm{d}}{\mathrm{d}x}$ can be played by $\delta_{x}$, the central-difference operator in the imaginary direction (§18.1(i)). This happens, for example, with the continuous Hahn polynomials and Meixner–Pollaczek polynomials (§18.20(i)).

## §18.2(iii) Normalization

The orthogonality relations (18.2.1)–(18.2.3) each determine the polynomials $p_{n}(x)$ uniquely up to constant factors, which may be fixed by suitable normalization.

If we define

 18.2.5 $h_{n}=\int_{a}^{b}\left(p_{n}(x)\right)^{2}w(x)\,\mathrm{d}x\text{ or }\sum_{x% \in X}\left(p_{n}(x)\right)^{2}w_{x},$ ⓘ Defines: $h_{n}$ (locally) Symbols: $\,\mathrm{d}\NVar{x}$: differential, $\in$: element of, $\int$: integral, $w(x)$: weight function, $w_{x}$: weights, $X$: subset of $\mathbb{R}$, $n$: nonnegative integer, $p_{n}(x)$: polynomial of degree $n$ and $x$: real variable A&S Ref: 22.1.2 Referenced by: §3.5(v) Permalink: http://dlmf.nist.gov/18.2.E5 Encodings: TeX, pMML, png See also: Annotations for §18.2(iii), §18.2 and Ch.18
 18.2.6 $\tilde{h}_{n}=\int_{a}^{b}x\left(p_{n}(x)\right)^{2}w(x)\,\mathrm{d}x\text{ or% }\sum_{x\in X}x\left(p_{n}(x)\right)^{2}w_{x},$

and

 18.2.7 $p_{n}(x)=k_{n}x^{n}+\tilde{k}_{n}x^{n-1}+\tilde{\tilde{k}}_{n}x^{n-2}+\cdots,$ ⓘ Defines: $k_{n}$ (locally), $\tilde{k}_{n}$ (locally) and $\tilde{\tilde{k}}_{n}$ (locally) Symbols: $n$: nonnegative integer, $p_{n}(x)$: polynomial of degree $n$ and $x$: real variable A&S Ref: 22.1.2 Referenced by: §18.15(vi), §3.5(v) Permalink: http://dlmf.nist.gov/18.2.E7 Encodings: TeX, pMML, png See also: Annotations for §18.2(iii), §18.2 and Ch.18

then two special normalizations are: (i) orthonormal OP’s: $h_{n}=1$, $k_{n}>0$; (ii) monic OP’s: $k_{n}=1$.

## §18.2(iv) Recurrence Relations

As in §18.1(i) we assume that $p_{-1}(x)\equiv 0$.

### First Form

 18.2.8 $p_{n+1}(x)=(A_{n}x+B_{n})p_{n}(x)-C_{n}p_{n-1}(x),$ $n\geq 0$. ⓘ Symbols: $n$: nonnegative integer, $p_{n}(x)$: polynomial of degree $n$, $B_{n}$: real constant, $C_{n}$: real constant, $x$: real variable and $A_{n}$: real constant A&S Ref: 22.1.4 Referenced by: §18.30 Permalink: http://dlmf.nist.gov/18.2.E8 Encodings: TeX, pMML, png See also: Annotations for §18.2(iv), §18.2(iv), §18.2 and Ch.18

Here $A_{n}$, $B_{n}$ ($n\geq 0$), and $C_{n}$ ($n\geq 1$) are real constants, and $A_{n-1}A_{n}C_{n}>0$ for $n\geq 1$. Then

 18.2.9 $\displaystyle A_{n}$ $\displaystyle=\frac{k_{n+1}}{k_{n}},$ $\displaystyle B_{n}$ $\displaystyle=\left(\frac{\tilde{k}_{n+1}}{k_{n+1}}-\frac{\tilde{k}_{n}}{k_{n}% }\right)A_{n}=-\frac{\tilde{h}_{n}}{h_{n}}A_{n},$ $\displaystyle C_{n}$ $\displaystyle=\frac{A_{n}\tilde{\tilde{k}}_{n}+B_{n}\tilde{k}_{n}-\tilde{% \tilde{k}}_{n+1}}{k_{n-1}}=\frac{A_{n}}{A_{n-1}}\frac{h_{n}}{h_{n-1}}.$ ⓘ Defines: $B_{n}$: real constant (locally), $C_{n}$: real constant (locally) and $A_{n}$: real constant (locally) Symbols: $n$: nonnegative integer, $h_{n}$, $\tilde{h}_{n}$, $k_{n}$, $\tilde{k}_{n}$ and $\tilde{\tilde{k}}_{n}$ A&S Ref: 22.1.5 Permalink: http://dlmf.nist.gov/18.2.E9 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §18.2(iv), §18.2(iv), §18.2 and Ch.18

### Second Form

 18.2.10 $xp_{n}(x)=a_{n}p_{n+1}(x)+b_{n}p_{n}(x)+c_{n}p_{n-1}(x),$ $n\geq 0$.

Here $a_{n}$, $b_{n}$ ($n\geq 0$), $c_{n}$ ($n\geq 1$) are real constants, and $a_{n-1}c_{n}>0$ ($n\geq 1$). Then

 18.2.11 $\displaystyle a_{n}$ $\displaystyle=\frac{k_{n}}{k_{n+1}},$ $\displaystyle b_{n}$ $\displaystyle=\frac{\tilde{k}_{n}}{k_{n}}-\frac{\tilde{k}_{n+1}}{k_{n+1}}=% \frac{\tilde{h}_{n}}{h_{n}},$ $\displaystyle c_{n}$ $\displaystyle=\frac{\tilde{\tilde{k}}_{n}-a_{n}\tilde{\tilde{k}}_{n+1}-b_{n}% \tilde{k}_{n}}{k_{n-1}}=a_{n-1}\frac{h_{n}}{h_{n-1}}.$ ⓘ Defines: $a_{n}$: real constant (locally), $b_{n}$: real constant (locally) and $c_{n}$: real constant (locally) Symbols: $n$: nonnegative integer, $h_{n}$, $\tilde{h}_{n}$, $k_{n}$, $\tilde{k}_{n}$ and $\tilde{\tilde{k}}_{n}$ Permalink: http://dlmf.nist.gov/18.2.E11 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §18.2(iv), §18.2(iv), §18.2 and Ch.18

If the OP’s are orthonormal, then $c_{n}=a_{n-1}$ ($n\geq 1$). If the OP’s are monic, then $a_{n}=1$ ($n\geq 0$).

Conversely, if a system of polynomials $\{p_{n}(x)\}$ satisfies (18.2.10) with $a_{n-1}c_{n}>0$ ($n\geq 1$), then $\{p_{n}(x)\}$ is orthogonal with respect to some positive measure on $\mathbb{R}$ (Favard’s theorem). The measure is not necessarily of the form $w(x)\,\mathrm{d}x$ nor is it necessarily unique.

## §18.2(v) Christoffel–Darboux Formula

 18.2.12 $\sum_{\ell=0}^{n}\frac{p_{\ell}(x)p_{\ell}(y)}{h_{\ell}}=\frac{k_{n}}{h_{n}k_{% n+1}}\frac{p_{n+1}(x)p_{n}(y)-p_{n}(x)p_{n+1}(y)}{x-y},$ $x\neq y$.

### Confluent Form

 18.2.13 $\sum_{\ell=0}^{n}\frac{(p_{\ell}(x))^{2}}{h_{\ell}}=\frac{k_{n}}{h_{n}k_{n+1}}% {\left(p_{n+1}^{\prime}(x)p_{n}(x)-p_{n}^{\prime}(x)p_{n+1}(x)\right)}.$

## §18.2(vi) Zeros

All $n$ zeros of an OP $p_{n}(x)$ are simple, and they are located in the interval of orthogonality $(a,b)$. The zeros of $p_{n}(x)$ and $p_{n+1}(x)$ separate each other, and if $m then between any two zeros of $p_{m}(x)$ there is at least one zero of $p_{n}(x)$.

For illustrations of these properties see Figures 18.4.118.4.7.